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On the unbiased asymptotic normality of quantile regression with fixed effects
abstract\begin{spacing}{1}
Nonlinear panel data models with fixed individual effects provide an important set of tools for describing microeconometric data. In a large class of such models (including probit, proportional hazard and quantile regression to name just a few) it is impossible to difference out the individual effects, and inference is usually justified in a `large $n$ large $T$' asymptotic framework. However, there is a considerable gap in the type of assumptions that are currently imposed in models with smooth score functions (such as probit, and proportional hazard) and quantile regression. In the present paper we show that this gap can be bridged and
establish unbiased asymptotic normality for fixed effects quantile regression panels under conditions on $n,T$ that are very close to what is typically assumed in standard nonlinear panels. Our results considerably improve upon existing theory and show that quantile regression is applicable to the same type of panel data (in terms of $n,T$) as other commonly used nonlinear panel data models. Thorough numerical experiments confirm our theoretical findings.
\end{spacing}
\thispagestyle{empty}
Keywords: Quantile regression; panel data; fixed effects; asymptotics
\newline
JEL Classification: C13, C21, and C23
\baselineskip18.9pt \pagenumbering{arabic}
Introduction
Nonlinear panel data models are important in microeconometric applications. They provide flexible modeling tools and allow researchers to account for unobserved heterogeneity through individual specific effects. Examples of such models include, among others, probit, logit, Poisson, negative binomial, proportional hazard, tobit, and quantile regression. Nevertheless, incorporating individual-specific fixed effects (FE) into these frameworks results in models whose dimension depends on the number of individuals, and thus grows with sample size; this raises some important issues for the asymptotic analysis of the FE estimators. As first noted by NeymanScott48, leaving the individual heterogeneity unrestricted in a nonlinear or dynamic panel model can result in inconsistent estimators of the common parameters due to the incidental parameters problem. Consistency can be recovered in settings where it is possible to remove individual-specific effects by a transformation or when the number of individuals, $n$, and the number of time periods, $T$, jointly go to infinity.
The nonlinear panel data literature has shown that $n/T \to 0$ is a sufficient condition to obtain asymptotic (unbiased) normality of nonlinear panel data FE estimators under smoothness conditions on the objective function.\footnote{Under an asymptotic framework where both the numbers of individuals and time periods grow at the same rate, it is possible to show that the fixed effects estimator for smooth objective functions has a limiting normal distribution with non-zero mean.} We refer the reader to ArellanoHahn07, ArellanoBonhomme11, and Fernandez-ValWeidner17 for reviews of the literature.\footnote{Important work in this literature include, among others, Kiviet95, PhillipsMoon99, Lancaster00, HahnKuersteiner02, Lancaster02, Arellano03a, AlvarezArellano03, LiLindsayWaterman03, HahnNewey04, Woutersen04, Carro07, BesterHansen09, Fernandez-Val09, HahnKuersteiner11, Fernandez-ValLee13, DhaeneJochmans15, Fernandez-ValWeidner16.}
An important class of nonlinear panel data models is quantile regression (QR). Panel QR models have provided a valuable method of statistical analysis and recent examples of their application include instrumental variables models [e.g., ChetverikovLarsenPalmer16, Galvao11, and HardingLamarche09], nonseparable models with time homogeneity [ChernozhukovFernandez-ValHoderleinHolzmannNewey15, and ChernozhukovFernandez-ValHahnNewey13], censored regression models [GalvaoLamarcheLima2013, and WangFygenson09], nonlinear models [ArellanoBonhomme16], interactive effects [HardingLamarche14], and growth charts [WeiHe06], amongst many others.\footnote{For other recent developments in panel data QR, see e.g., Canay11, GrahamHahnPoirierPowell2015, SuHoshino16, GalvaoKato16, and HardingLamarche17. GalvaoKato17 review the QR methods for panel (longitudinal) data.}
Unfortunately, as in most of the nonlinear panel data literature, transformations to remove the unobserved individual effects are not available for QR models. Thus, FE panel data QR suffers from the incidental parameters problem and large $n,T$ asymptotics must be employed in the analysis. KatoGalvaoMontes-Rojas12 analyze the standard fixed effects (FE-QR) estimator in which individual effects are introduced through dummy variables, and GalvaoWang15 consider the minimum distance (MD-QR) estimator. These studies establish the asymptotic properties of the corresponding estimators and derive sufficient conditions for their consistency and asymptotic normality. Both the FE-QR and MD-QR estimators are shown to be asymptotically normal under the stringent condition that $n^{2}(\log n)(\log T)^2/T\to0$, and $T \to \infty$ as $n \to \infty$. This requirement is much more restrictive than what is usually required in the standard literature on nonlinear panel data models under smoothness conditions.
The substantial discrepancy between existing conditions that guarantee asymptotic normality of panel data QR and other nonlinear panel models gives rise to the following question: are the restrictive conditions imposed in the QR literature really necessary, or are these conditions rather an artifact of the proof techniques employed so far?\footnote{The source of this stronger requirement is the lack of smoothness of the QR objective function, which implies that classical higher-order expansions are not valid.} Answering this question is of central importance to the panel data QR literature and will have profound implications for the recommendation that econometricians can give to practitioners when it comes to the set of tools that should be used in the analysis of panel data with moderate length. If indeed the conditions required for QR turn out to be substantially more restrictive compared to other nonlinear models, it would substantially limit the application of QR to panels that have only modest time dimension compared to the cross sectional dimension.
The main contribution of this paper is to provide an answer to the question posed above. We prove that unbiased asymptotic normality of both the FE-QR and MD-QR estimators continues to hold provided that $n(\log T)^{2}/T \to 0$, as $(n,T) \to \infty$. This significantly improves upon the previous condition available in the literature, $n^{2}(\log n)(\log T)^2/T\rightarrow0$, and shows that QR is applicable to the same type of panels as other nonlinear models. In addition, we provide a complete set of corresponding results under conditions that allow for temporal dependence of observations within individuals, thus encompassing a large class of possible empirical applications.
While the results that we obtain are close to what is established in most of the nonlinear panel data literature under smooth conditions, we would like to emphasize that the method of proof is rather different. The main difficulty stems from the non-smooth objective function which renders most of the usual techniques (which crucially rely on Taylor expansions of the objective function) inapplicable. A key insight in the proofs is a detailed analysis of the expected values of remainder terms in the classical Bahadur representation for QR, while previous approaches (including KatoGalvaoMontes-Rojas12 and GalvaoWang15) focused on the stochastic order of those remainder terms. A similar analysis was previously performed in VCC for general QR models with growing dimension under the assumption of independent and identically distributed observations. The proofs involve subtle empirical process arguments, and extending those results to settings with dependent data requires a substantial amount of work. We further note that the results in VCC are not directly applicable to either of the estimators we consider here. The FE-QR estimator can not be expressed as an average at all, and several intermediate steps in the Bahadur representation are needed before the ideas from VCC become applicable. The MD-QR estimator is itself an average, but a complication in the corresponding analysis is due to the estimated weights, which also depend on estimates of the quantile coefficients in a non-smooth way.
Finally, we conduct Monte Carlo simulations to study the finite sample properties of the FE-QR and MD-QR estimators and verify our theory. The numerical experiments confirm the theoretical findings.
The rest of the paper is organized as follows. Section (ref) describes the QR model and the estimators. Section (ref) contains the asymptotic results, theory in the independent case is presented in Section (ref) while Section (ref) considers extensions to temporal dependence within individuals. In Section (ref) we present the Monte Carlo experiments and conclude in Section (ref). All proofs are collected in the Appendix.
The model and estimators
The model
Assume that we have observations $(Y_{it},{\mathbf{X}}_{it})_{i=1,..,n, t=1,...,T}$ where $Y_{it}$ denotes the response variable for individual $i$ at observation period $t$, and ${\mathbf{X}}_{it}$ denotes the corresponding $p$-dimensional vector of explanatory variables. In this paper, we use a quantile regression (QR) panel data model with fixed effects (FE) to describe the influence of ${\mathbf{X}}_{it}$ on the response $Y_{it}$. This model takes the form
equation[equation omitted — 151 chars of source]
where $q_{i,\tau}({\mathbf{X}}_{it})=\inf\{Y:\text{Pr}(Y_{it}<Y|{\mathbf{X}}_{it})\geq\tau\}$ is the conditional $\tau$-quantile of $Y_{it}$ given ${\mathbf{X}}_{it}$. Here $\bm{\beta}_0(\tau)$ denotes the vector of common slope coefficients while $\alpha_{i0}(\tau)$ is a quantile-specific individual effect, which is intended to capture individual specific sources of variability, or unobserved heterogeneity that was not adequately controlled by other covariates in the model. In what follows, we will also use the notation $\bm{\gamma}_{i0}(\tau)=(\alpha_{i0}(\tau),\bm{\beta}_{0}(\tau)^{\top})^{\top}$ and ${\mathbf{Z}}_{it}^{\top}=(1,{\mathbf{X}}_{it}^{\top})$. This model is semiparametric in the sense that the functional form of the conditional distribution of $Y_{it}$ given ${\mathbf{X}}_{it}$ is left unspecified and no parametric assumption is made on the relation between ${\mathbf{X}}_{it}$ and $\alpha_{i0}$.
Next we describe two estimators for the parameters of interest, $\bm{\beta}_{0}(\tau)$, namely the standard fixed effects QR and the minimum distance QR.
The estimators
To estimate the panel QR model, Koenker04 considers the fixed effects quantile regression (FE-QR) estimator. This is an individual dummy variables estimator, which is a natural analog of the dummy variables estimator for the standard FE mean regression model. However, in contrast to mean regression, there is no general transformation that can suitably eliminate the individual specific effects in the FE-QR estimator, and hence one is required to deal directly with the full problem. The FE-QR is defined as follows\footnote{We follow Koenker04 in treating the $\alpha_i$ as fixed parameters. An alternative approach which leads to equivalent results is to treat the $\alpha_i$ as random (with no restrictions placed on the dependence with $X_{it}$). In this case the model can be written as $Q_{Y_{it}|X_{it},\alpha_i(\tau)}(\tau) = \beta_0(\tau)^\top x + \alpha_{0i}(\tau)$; here $Q_{Y_{it}|X_{it},\alpha_i(\tau)}$ denotes the conditional quantile function of $Y_{it}$ given $(X_{it},\alpha_i(\tau))$ (see for instance KatoGalvaoMontes-Rojas12, GalvaoWang15 and GalvaoKato16 for this interpretation). Both interpretations lead to the same asymptotic results.}
equation[equation omitted — 297 chars of source]
where $\bm{\alpha} := (\alpha_{1},\dots,\alpha_{n})^{\top}$ and $\rho_{\tau}(u) := \{ \tau - \bm{1}(u \leq 0) \} u$ is the check function as in KoenkerBassett78. We refer to the FE-QR estimator defined in equation (ref) as $\widehat{\boldsymbol{\beta}}_{FE}(\tau)$.
In typical applications, the number of individuals can be large and the FE-QR estimator involves optimization with substantial number of parameters to be estimated, which makes the problem computationally cumbersome. Hence, motivated by the practical implementation challenges of the FE-QR, we also consider a simple to implement and efficient QR minimum distance (MD) estimator for panels with fixed effects to estimate the common parameter of interest $\bm{\beta}_{0}(\tau)$.\footnote{The MD estimation is a flexible methodology and has been largely applied to panel data problems, examples, among others, include Swamy70, Chamberlain82, Chamberlain84, AhnSchmidt95, HsiaoPesaranTahmiscioglu02, Hsiao03, Pesaran06, LeeMoonWeidner12, and MoonShumWeidner17.} As in GalvaoWang15, we use a minimum distance quantile regression (MD-QR) estimator, $\widehat{\bm{\beta}}_{MD}^{\rm{inf}}(\tau)$, defined as follows
equation[equation omitted — 166 chars of source]
where $\widehat{\bm{\beta}}_{i}(\tau)$ is the slope coefficient estimator from each individual QR problem using the time-series data, i.e.
equation*[equation* omitted — 306 chars of source]
where $W_i$ denotes the associated variance-covariance matrix of $\widehat{\bm{\beta}}_{i}(\tau)$ for each individual.
However, in applications, the estimator $\widehat{\boldsymbol{\beta}}^{\rm{inf}}_{MD}$, defined in equation (ref), is infeasible unless $W_i$ is known for every individual. The feasible estimator is defined with each $W_i$ replaced by its corresponding consistent estimators $\widehat{W}_i$, such that the MD-QR is given by
equation[equation omitted — 180 chars of source]
This feasible two-step estimator can be implemented by obtaining, in the first step, consistent estimates for the slope coefficients and their associated variance-covariance matrices, $\widehat{W}_{iT}$. One can obtain such estimates from the standard QR algorithm for each individual separately. In the second step the estimated $W_{i}$'s are substituted into (ref) which results in (ref). The specific form of $\widehat{W}_{iT}$ depends on the assumption on the dependence across individuals, detailed expressions for such estimators will be provided later.
Asymptotic theory
{In this section, we investigate the asymptotic properties of the FE-QR estimator in (ref) and also the MD-QR estimator in (ref). We begin by stating and discussing a set of basic assumptions that will be used throughout the remaining parts of this paper. In Section (ref) we discuss the asymptotic distribution of the FE-QR and the feasible MD-QR estimators given in (ref) and (ref), respectively, under the additional assumption that data within individuals are independent. An extension of those results to the case where temporal dependence of observations within individuals is allowed is provided in Section (ref). }
Basic assumptions
Throughout the paper, we use the following notations: for a square matrix $A$, let $\|A\|$ denote the maximal absolute eigenvalue of $A$, while for a vector $v$, $\|v\|$ denotes the usual Euclidean norm. Recall that ${\mathbf{Z}}_{it}^\top = (1, {\mathbf{X}}_{it}^\top)$ is a vector of dimension $p+1$ and let $\mathcal{Z}$ denote the support of ${\mathbf{Z}}_{it}$. We make the following assumptions.
itemize• The processes $\{ (Y_{it},{\mathbf{X}}_{it}) : t \in \mathbb{Z} \}$ are strictly stationary for each $i$ and independent across $i$.
• Assume that $\|{\mathbf{Z}}_{it}\| \leq M < \infty$ almost surely, and that $c_{\lambda}\leq\lambda_{\min}({\mathbb{E}} [{\mathbf{Z}}_{it} {\mathbf{Z}}_{it}^\top])\leq\lambda_{\max}({\mathbb{E}} [{\mathbf{Z}}_{it} {\mathbf{Z}}_{it}^\top])\leq C_{\lambda}$ holds uniformly in $i$ for some fixed constants $c_{\lambda}>0$ and $C_{\lambda} <\infty$.
• The conditional distribution $F_{Y_{i1}|{\mathbf{Z}}_{i1}}(y|\mathbf{z})$ is twice differentiable w.r.t. $y$, with the corresponding derivatives $f_{Y_{i1}|{\mathbf{Z}}_{i1}}(y|\mathbf{z})$ and $f'_{Y_{i1}|{\mathbf{Z}}_{i1}}(y|\mathbf{z})$. Assume that
\begin{equation*}
f_{max} :=\sup_i \sup_{y \in \mathbb{R},\mathbf{z}\in \mathcal{Z}} |f_{Y_{i1}|{\mathbf{Z}}_{i1}}(y|\mathbf{z})| < \infty,
\end{equation*}
and
\begin{equation*}
\overline{f'} := \sup_i \sup_{y \in \mathbb{R},\mathbf{z}\in \mathcal{Z}} |f'_{Y_{i1}|{\mathbf{Z}}_{i1}}(y|\mathbf{z})| < \infty.
\end{equation*}
• Denote by $\mathcal{T}$ an open neighborhood of $\tau$. Assume that uniformly across $i$, there exists a constant $f_{\min} < f_{\max}$ such that
\begin{equation*}
0 < f_{\min} \leq \inf_i \inf_{\eta \in \mathcal{T}} \inf_{\mathbf{z} \in \mathcal{Z}} f_{Y_{i1}|{\mathbf{Z}}_{i1}}(q_{i,\eta}(\mathbf{z})|\mathbf{z}),
\end{equation*}
where $q_{i,\eta}(\mathbf{z})$ is the conditional $\eta$ quantile of $Y_{i1}$ given ${\mathbf{Z}}_{i1} = \mathbf{z}$.
Condition (A0) assumes that the data are independent across individuals, and strictly stationary within each individual. Additional assumptions on the temporal dependence within individuals will be added later when we state the corresponding results. Condition (A1) poses a boundedness condition on the norm of the regressors, which is also standard in the literature, see for instance Koenker04, KatoGalvaoMontes-Rojas12, GalvaoWang15. Condition (A1) also assures that the eigenvalues of ${\mathbb{E}} [{\mathbf{Z}}_{it} {\mathbf{Z}}_{it}^\top]$ are bounded away from zero and infinity uniformly across $i$, similar assumptions were made in CVC, BCCF. Conditions (A2) and (A3) impose smoothness and boundedness of the conditional distribution, the density and its derivatives. The same type of assumption has been imposed in GalvaoWang15. CVC and BCCF also make similar assumptions when deriving Bahadur representations for QR estimators in a setting without panel data.
The independent case
The main contribution of this section is to provide new insights on the asymptotic properties of the FE-QR estimator in (ref) and the feasible version of the MD-QR estimator in (ref) under the assumption that data are independent and identically distributed (i.i.d.) within individuals. We are able to considerably improve the existing conditions on $n,T$, thus reconciling asymptotic results in the panel data QR and nonlinear panel literature. Throughout this section, we will use the following assumption.
enumerate• Assume that for each $i$, the observations are $({\mathbf{X}}_{it},Y_{it})_{t=1,...,T}$ are i.i.d. across $t$. Moreover, assume that $n \to \infty, T \to \infty$ and\footnote{The assumption $n \to \infty$ could be dropped at the cost of more complicated notation. Nevertheless, since this case can be handled by existing results we prefer to concentrate on the more complicated setting $n \to \infty$. In addition, we conjecture that the factor $(\log T)^2$ could be improved, but that seems impossible with our current method of proof. We leave such an improvement for future research.}
\begin{equation*}
\frac{n (\log T)^2}{T} = o(1).
\end{equation*}
Condition (I) excludes temporal dependence. In Section (ref), we extend the results to the dependent case under suitable mixing conditions as in HahnKuersteiner11. We note that the independence assumption is used mainly to apply some standard stochastic inequalities; our results are extended in Section (ref) to the dependent case by replacing these stochastic inequalities by those that hold under suitable dependence conditions.
We begin by discussing the FE-QR estimator. To this end let $f_i(y) := {\mathbb{E}}[f_{Y_{i1}|{\mathbf{X}}_{i1}}(y+q_{i,\tau}({\mathbf{X}}_{i1})|{\mathbf{X}}_{i1})]$ denote the marginal density of $Y_{i1} - q_{i,\tau}({\mathbf{X}}_{i1})$ and define
equation[equation omitted — 157 chars of source]
Let
equation[equation omitted — 211 chars of source]
enumerate• Assume that $\Gamma_n$ is non-singular for each $n$, that $\Gamma := \lim_{n \to \infty} \Gamma_n$ exists and is non-singular and that $L := \lim_{n \to \infty} n^{-1}\sum_{i=1}^n {\mathbb{E}}[({\mathbf{X}}_{i1}-g_i)({\mathbf{X}}_{i1}-g_i)^\top]$ exists and is non-singular.
theoremAssume that (A0)--(A3), (I), (FI) hold. Then, for fixed $\tau\in(0,1)$, we have that
\begin{equation}
\sqrt{nT}(\widehat{\bm{\beta}}_{FE}(\tau) - \bm{\beta}_{0}(\tau)) \stackrel{\mathcal{D}}{\longrightarrow} \mathcal{N}\big(0,\tau(1-\tau)\Gamma^{-1}L\Gamma^{-1}\big).
\end{equation}
The limiting distribution above is the same as in Theorem 3.2 of KatoGalvaoMontes-Rojas12. However, in contrast to the assumption $n^2(\log n)^3/T = o(1)$ in the latter reference we only require $n (\log T)^2/T = o(1)$. Some intuition on why such an improvement is possible is provided in Remark (ref) below. Finally, note that consistent estimation of the asymptotic variance-covariance matrix $\tau(1-\tau)\Gamma^{-1}L\Gamma^{-1}$ was discussed in KatoGalvaoMontes-Rojas12, see their Proposition 3.1. Since that result does not require a stringent growth condition on $n$ it continues to be applicable under our weaker assumptions.
We next discuss the MD-QR estimator. To complete the computation of the MD-QR estimator in (ref) we need an estimate $\widehat{W}_{iT}$. Assuming that data within individuals are i.i.d., the asymptotic covariance matrix of $\widehat{\bm{\gamma}}_{i}(\tau)=(\widehat{\alpha}_{i}(\tau),\widehat{\bm{\beta}}_{i}(\tau))$ takes the form $V_i = B_i^{-1}A_iB_i^{-1}$ where
equation[equation omitted — 249 chars of source]
{Note that $W_{i}$ is the lower $p\times p$ sub-matrix of $V_{i}$. Hence by estimating $V_{i}$ consistently we recover a consistent estimator of $W_{i}$.} A common way to estimate $V_{i}$ uses the Hendricks-Koenker sandwich covariance matrix estimator (HendricksKoenker91) which takes the following form
equation[equation omitted — 114 chars of source]
with
align*[align* omitted — 387 chars of source]
Here $d_T$ denotes a smoothing parameter that should converge to zero at an appropriate rate in order to guarantee consistency of $\widehat{B}_{iT}$, which is imposed in assumption (MI) below.
Given the definitions above, we consider the following MD-QR estimator
equation[equation omitted — 172 chars of source]
where $\widehat{W}_{iT}$ is the lower $p\times p$ sub-matrix of $\widehat{V}_{iT}$ in (ref). Since consistency of a closely related estimator (the only difference is the form of the variance estimators $\widehat{W}_{iT}$) was established in GalvaoWang15, we focus on conditions which ensure asymptotic normality. {In addition to (A0)-(A3), (I) we make the following assumption
enumerate• Assume that $d_T = o(1)$, as $T \to \infty$ and
\begin{equation*}
\frac{\log T}{T d_T^2} = o(1).
\end{equation*}
Let $W_i$ denote the lower $p\times p$ sub-matrix of $V_i$ and assume that $\left( \frac{1}{n}\sum_{i=1}^{n} W_i^{-1} \right)^{-1}$
is a well-defined positive definite matrix if $n$ is fixed and that $\Sigma_{1} := \lim_{n\to \infty} (\frac{1}{n}\sum_{i=1}^{n} W_i^{-1})^{-1}$ exists and is a well-defined positive definite matrix if $n \to \infty$.
}
The asymptotic normality of the MD-QR estimator is formalized in the next theorem.
theoremAssume that (A0)--(A3), (I) and (MI) hold. In this case, for fixed $\tau$, we have that
\begin{equation}
\sqrt{nT}(\widehat{\bm{\beta}}_{MD}(\tau) - \bm{\beta}_{0}(\tau)) \stackrel{\mathcal{D}}{\longrightarrow} \mathcal{N}\big(0,\Sigma_{1}\big).
\end{equation}
An important by-product of the proof of Theorem (ref) is that we are able to estimate the asymptotic variance-covariance matrix, $\Sigma_{1}$, consistently, uniformly over a growing number of individuals. More precisely, we obtain the following result.
lemmaUnder the conditions of Theorem (ref), we have $\max_{i=1,...,n}\|\widehat{B}_{iT} - B_{i} \| = o_P(1)$ and $\max_{i=1,...,n}\|\widehat{A}_{iT} - A_{i} \| = o_P(1)$. Moreover, $\widehat{\Sigma}_1 := (\frac{1}{n}\sum_{i=1}^n \widehat{W}_{iT}^{-1} )^{-1}$ is a consistent estimator for $\Sigma_1$.
The result in Lemma (ref) allows for the construction of simple to implement inference procedures based on the Wald statistic.
remark{\rm In the proof of Theorem (ref) in the Appendix (see Lemma (ref)), we also provide uniform convergence rates and Bahadur representations of the estimators $\widehat{A}_{iT},\widehat{B}_{iT}$. Those results directly yield a corresponding uniform rate and Bahadur representation for the Hendricks-Koenker covariance matrix estimator $\widehat{V}_{it}$. This result may be of independent interest. }
The following remarks describe the intuition for the proofs of Theorems (ref) and (ref). Since the proof of Theorem (ref) builds on intermediate results as well as intuition from the proof of Theorem (ref), we begin by discussing some of the key ideas in the proof of the latter result, which, to the best of our knowledge, are not present in the existing literature.
remark[Intuition for proofs of the MD estimator] {\rm
Theorem (ref) shows asymptotic normality of $\widehat{\bm{\beta}}_{MD}(\tau)$ under the assumption $n (\log T)^2/T = o(1)$, which considerably relaxes previous conditions of the form $n^2 (\log n)(\log T)^2/T = o(1)$ (see GalvaoWang15). The crucial insight for providing the improved condition comes from a closer analysis of the remainder term in the Bahadur representation for $\widehat{{\bm{\beta}}}_{MD}$
\begin{equation}
\widehat{\bm{\beta}}_{MD}(\tau) - \bm{\beta}_{0}(\tau) = \frac{1}{n} \sum_{i=1}^{n} W_{i}^{-1} \frac{1}{T}\sum_{t=1}^{T} \phi_{i,\tau}({\mathbf{Z}}_{it}, Y_{it}) + \frac{1}{n} \sum_{i=1}^{n} r_{n,i}(\tau) + o_p\Big(\frac{1}{\sqrt{nT}}\Big),
\end{equation}
where $\phi_{i,\tau}({\mathbf{Z}}_{it}, Y_{it})$ denotes the last $p$ entries of $B_i^{-1} {\mathbf{Z}}_{it} (\1(Y_{it} \leq q_{i,\tau}({\mathbf{Z}}_{it}))-\tau)$. Previous approaches bound the sum of remainder terms $\frac{1}{n} \sum_{i=1}^{n} r_{n,i}(\tau)$ by $\sup_i|r_{n,i}(\tau)|$, which is of order $O_P((\log T)^b/T^{3/4})$ for some constant $b$. The power of $T$ in this bound cannot be improved, which seems to suggest that in order to obtain unbiased asymptotic normality the condition $\sup_i|r_{n,i}(\tau)| = o_P((nT)^{-1/2})$ needs to be imposed. This gave rise to the condition $n^2 (\log n)(\log T)^2/T = o(1)$ in GalvaoWang15.
The intuitive reason for why this approach results in conditions that are too strong is that the remainder terms $r_{n,i}$ can be viewed as independent (across $i$) random vectors, and thus the order of their mean is governed by their expected values, while the variance is of the order $n^{-1}\sup_i Var(r_{n,i}(\tau)) = o((nT)^{-1/2})$. In order to make this intuition precise, we need to derive a more detailed expansion for $\widehat{\bm{\beta}}_{MD}(\tau) - \bm{\beta}_{0}(\tau)$ which allows to bound the expected values of the remainder terms. To achieve this, we start by deriving a Bahadur type expansion for the estimated weights which takes the form (see equation (ref) in the Appendix)
\begin{equation*}
\widehat{W}_{iT}^{-1} - W_{iT}^{-1} = \frac{1}{T} \sum_{t} \tilde{\eta}_{iT}({\mathbf{Z}}_{it}, Y_{it}) + \tilde{R}_{3i}(\tau),
\end{equation*}
where $\tilde{\eta}_{iT}({\mathbf{Z}}_{it}, Y_{it})$ are centered random variables with $(\tilde{\eta}_{iT}({\mathbf{Z}}_{it}, Y_{it}))_{t=1,...,T}$ i.i.d. for all $i$ and the remainder terms $\tilde{R}_{3i}(\tau)$ are small in a suitable sense. A similar expansion is derived for the estimators $\widehat{\bm{\beta}}_i$ which is given by (see Lemma (ref) in the Appendix)
\begin{equation*}
\widehat{\bm{\beta}}_{i}(\tau) - \bm{\beta}_{0}(\tau) = \frac{1}{T}\sum_{t=1}^{T} \phi_{iT}({\mathbf{Z}}_{it}, Y_{it}) + R_{iT}^{(1)}(\tau) + R_{iT}^{(2)}(\tau),
\end{equation*}
where again $\phi_{iT}({\mathbf{Z}}_{it}, Y_{it})$ are i.i.d. and centered, $R_{iT}^{(2)}(\tau)$ are `small' uniformly, and ${\mathbb{E}}[R_{iT}^{(1)}(\tau)]$ is `small'. Combining the two representations above yields the representation in (ref), and a detailed analysis of all remainder terms shows that, in fact, $\sup_i \Big\| {\mathbb{E}}[r_{n,i}(\tau)] \Big\| = O(\log T/T)$, which gives rise to the less restrictive condition $\log T/T = o((nT)^{-1/2})$.
We conclude this remark by noting that the idea of analyzing expected values of remainder terms when aggregating QR from subsets was also used in VCC. However, the setting we consider here is different from the latter paper since we also need to take into account the $\widehat{W}_{iT}^{-1}$ which also depend on $\widehat{\bm{\beta}}_{i}(\tau \pm d_T)$ in a non-smooth way. This considerably complicates the asymptotic analysis.
}
{
remark[Intuition for proofs of the FE estimator] {\rm Similarly to the discussion of the MD estimator given above, the main improvement in the conditions for the FE estimator comes from a closer analysis of the remainder terms in the Bahadur representation of $\widehat{\bm{\beta}}_{FE}(\tau)$. However, the structure of the estimator is quite different and so a different analysis is required. More precisely, a closer look at the proof of Theorem 3.2 in KatoGalvaoMontes-Rojas12 shows that the remainder term which is responsible for the condition $n(\log n)^2/T = o(1)$ in the latter paper is of the form
\[
\frac{1}{n} \sum_{i=1}^n \mathbb{H}_{ni}^{(3)}(\widehat\alpha_i,\widehat{\bm{\beta}}) - \mathbb{H}_{ni}^{(3)}(\alpha_{i0},{\bm{\beta}}_0) - \Big\{H_{ni}^{(3)}(\widehat\alpha_{i},\widehat{\bm{\beta}}) - H_{ni}^{(3)}(\alpha_{i0},{\bm{\beta}}_0)\Big\},
\]
where $\mathbb{H}_{ni}^{(3)} - H_{ni}^{(3)}$ denotes certain empirical processes with each process using only observations from individual $i$ (see the beginning of section A.2 for a formal definition). A direct analysis of this remainder in KatoGalvaoMontes-Rojas12 leads to a stochastic order $O_P((\log(n)/T)^{3/4})$ which can not be improved (in terms of the power of $T$) and gives rise to the condition $n^2(\log n)^3/T = o(1)$ in the latter paper. Given the discussion in the MD case it would seem natural to again look at the expected value of this remainder term. However, this is not directly helpful since the terms in the sum are dependent across $i$ through the common $\widehat {\bm{\beta}}$ and the $\widehat{\alpha}_i$ so that a direct control of the variance of this term is infeasible. The key idea to solve those difficulties is to first show that this remainder term is sufficiently close to
\[
\frac{1}{n} \sum_{i=1}^n \mathbb{H}_{ni}^{(3)}(\widetilde\alpha_i,{\bm{\beta}}_0) - \mathbb{H}_{ni}^{(3)}(\alpha_{i0},{\bm{\beta}}_0) - \Big\{H_{ni}^{(3)}(\widetilde\alpha_{i},{\bm{\beta}}_0) - H_{ni}^{(3)}(\alpha_{i0},{\bm{\beta}}_0)\Big\}
\]
where $\widetilde \alpha_i := \operatorname*{arg\,min}_{a \in R} \sum_{t=1}^T \rho_\tau(Y_{it} - {\mathbf{X}}_{it}^\top {\bm{\beta}}_0 - a)$. In contrast to the $\widehat\alpha_i$ which are dependent across $i$ because they result from running one single regression the $\widetilde\alpha_i$ only use data from individual $i$. Hence the terms in the above sum are now independent across $i$ and the ideas from VCC can be applied to show that their expected values are of order $T^{-1}\log T$ while the variance of the sum is negligible due to averaging. This eventually leads to the condition $\log T/T = o((nT)^{-1/2})$ which explains the source of our less restrictive growth condition.
}
}
The dependent case
In this section we extend the asymptotic results provided in Section (ref) to settings where we allow for dependence across time while still maintaining independence across individuals. We continue to use notations introduced in the previous section.
{ Throughout this section we will make the following assumptions on the temporal dependence structure.
itemize• For each $i \geq 1$, the process $\{ (Y_{it},{\mathbf{X}}_{it}) : t \in \mathbb{Z} \}$ is strictly stationary and $\beta$-mixing. Let $\beta_{i}(j)$ denote the $\beta$-mixing coefficient of the process $\{ (Y_{it},{\mathbf{X}}_{it}) :t \in \mathbb{Z} \}$. Assume that there exist constants $b_\beta \in (0,1), C_\beta > 0$ independent of $i$ such that
\begin{equation*}
\sup_i \beta_i(j) \leq C_{\beta} b_\beta^j =: \beta(j) \quad \forall j \geq 1.
\end{equation*}
• For each $i=1,...,n$ and $j>1$, the random vector $(Y_{i1},Y_{i 1+j})$ has a density conditional on $({\mathbf{Z}}_{i1},{\mathbf{Z}}_{i 1+j})$ and this density is bounded uniformly across $i,j$.
}
Condition (D1) relaxes the assumption of i.i.d. within each individual to that of stationary $\beta$-mixing which is used in KatoGalvaoMontes-Rojas12, GalvaoWang15 and is similar to HahnKuersteiner11. The $\beta$-mixing condition is stronger than $\alpha$-mixing. Nevertheless, $\beta$-mixing is still satisfied in a reasonably large class of time series models.\footnote{For example, consider the MA $(\infty)$ process $X_{t}= \sum_{j=0}^{ \infty}a_{j} \varepsilon_{t-j}$, where $a_{j} \to 0$ exponentially fast (note that the ARMA($p,q$) process, subject to standard assumptions, fulfils this condition), and $\{ \varepsilon_{t} \}$ is i.i.d. If the density function of $\varepsilon_{t}$ exists, then $\{ X_{t}\}$ is $\beta$-mixing with $\beta(n) \to 0$ exponentially fast.} Condition (D2) is needed because the data are not i.i.d. and we need to impose a condition on the joint distributions, {similar conditions were imposed in KatoGalvaoMontes-Rojas12, GalvaoWang15. }
{To state the asymptotic properties of $\widehat{\bm{\beta}}_{FE}(\tau)$ we make the following additional assumption.
enumerate• Assume that $\Gamma_n$ is non-singular for each $n$ and that $\lim_{n \to \infty} \Gamma_n$ exists and is non-singular. Further assume that
\[
\widetilde L := \lim_{n \to \infty} \frac{1}{n}\sum_{i=1}^n Var\Big(T^{-1/2}\sum_{t=1}^T ({\mathbf{X}}_{it}-g_i)(\tau - \1\{Y_{it} \leq {\mathbf{X}}_{it}^\top {\bm{\beta}}_0- \alpha_{i0}\})\Big)
\]
exists and is non-singular.
This assumption was also made by KatoGalvaoMontes-Rojas12 (see their condition (D3)), it involves the long run covariance matrix of the leading piece in the Bahadur representation for $\widehat {\bm{\beta}}_{FE}(\tau)$.
theoremAssume that (A0)--(A3), (D1)--(D2) and (FD) hold. If also $n (\log T)^4/T = o(1)$ then, for fixed $\tau$,
\begin{equation}
\sqrt{nT}(\widehat{\bm{\beta}}_{FE}(\tau) - \bm{\beta}_{0}(\tau)) \stackrel{\mathcal{D}}{\longrightarrow} \mathcal{N}\big(0,\tau(1-\tau)\Gamma^{-1}\widetilde L\Gamma^{-1}\big).
\end{equation}
Similarly as in the i.i.d. case we obtain the same limiting behaviour as KatoGalvaoMontes-Rojas12 but under a weaker condition on the growth rate of $n$ relative to $T$.
}
We next discuss the MD-QR estimator. In order to construct the MD-QR estimator with dependent data, we need to account for the dependence when estimating the covariance matrix. To this end we extend the Hendricks-Koenker estimator in the previous section. Note that the limiting variance of $\widehat{\bm{ \gamma}}_{i}$ in the dependent case takes the form $\widetilde V_{i} = B_{i}^{-1}\widetilde{A}_{i} B_{i}^{-1}$ where $B_i$ is the same as in the independent case (see (ref)) and
equation*[equation* omitted — 183 chars of source]
with $w_{it} := {\mathbf{Z}}_{it} (\tau - \1(Y_{it} \leq q_{i,\tau}(\bm{Z}_{it}))$. This motivates the following estimator of the limiting variance matrix of $\widehat{\bm{\gamma}}_{i}(\tau)$
equation[equation omitted — 116 chars of source]
where $\widehat{B}_{iT} := \frac{1}{T} \sum_{t=1}^{T} \widehat{f}_{it} {\mathbf{Z}}_{it}{\mathbf{Z}}_{it}^\top$ is defined in the same way as in the independent case and the estimator $\widetilde{A}_{iT}$ now takes the form
equation*[equation* omitted — 292 chars of source]
with $T_j := \{1\leq t \leq T - j\}$. Here $m_T$ is a `bandwidth' parameter (which needs to increase to infinity in order to obtain consistent estimation), and
equation*[equation* omitted — 135 chars of source]
The estimator $\widehat{\bm{\beta}}_{MD}(\tau)$ is defined as in (ref) with the new estimator $\widetilde{V}_{iT}$ in (ref) instead of $\widehat{V}_{iT}$.
To derive the asymptotic distribution of $\widehat{\bm{\beta}}_{MD}(\tau)$ we consider the following set of assumptions.
itemize• Assume that $d_T \to 0$, $m_T \to \infty$ and
\begin{equation*}
\frac{n (\log T)^4 m_T ^3}{T} = o(1), \quad \frac{\log T}{T d_T^2} = o(1).
\end{equation*}
Condition (MD) is similar to (MI) and imposes a restriction on the relative growth of the time dimension compared to number of individuals.
The following theorem provides the asymptotic normality result under stationary $\beta$-mixing dependence, and is an extension of Theorem (ref) in the previous section.
theoremAssume that (A0)--(A3) and (D1)--(D2) and (MD) hold and let $\widetilde W_{iT}$ denote the lower $p\times p$ sub-matrix of $\widetilde V_{iT}$. Assume that $\Sigma_{2}:= \lim_{n\to \infty} \Big(\frac{1}{n}\sum_{i=1}^{n} \widetilde W_{iT}^{-1}\Big)^{-1}$ exists and is a well-defined positive definite matrix. In this case, for fixed $\tau$, we have that
\begin{equation}
\sqrt{nT}(\widehat{\bm{\beta}}_{MD}(\tau) - \bm{\beta}_{0}(\tau)) \stackrel{\mathcal{D}}{\longrightarrow} N\big(0,\Sigma_{2} \big).
\end{equation}
Theorem (ref) shows that the asymptotic normality of the estimator for the dependent data still holds under similar conditions on $n,T$ as Theorem (ref). The following Lemma provides a result for the consistent estimation of the covariance matrix for the dependent case.
lemmaUnder the conditions of Theorem (ref), we have we have $\max_{i=1,...,n}\|\widehat{B}_{iT} - B_{i} \| = o_P(1)$ and $\max_{i=1,...,n}\|\widetilde{A}_{iT} - \widetilde A_{i} \| = o_P(1)$. Moreover, $\widehat \Sigma_2 := (\frac{1}{n}\sum_{i=1}^n \widetilde{W}_{iT}^{-1} )^{-1}$ is a consistent estimator for $\Sigma_2$.
As in the previous section, we can see that the asymptotic covariance matrix of $\widehat{\bm{\beta}}_{MD}(\tau)$ for the dependent case, $\Sigma_{2}$ is consistently estimated by the inverse of $\frac{1}{n} \sum_{i=1}^{n} \widetilde{W}_{iT}^{-1} $. The proof strategy is similar to the i.i.d. case (see Remark (ref) for an outline of the key steps) but somewhat more complicated due to the dependence within individuals. Note also that the estimators $\widetilde{A}_{iT}$ now contain an additional sum with $m_T \to \infty$ pieces, this poses an additional challenge for its theoretical analysis. As in the previous section, the result in Lemma (ref) allows for the construction of inference procedures based on the Wald statistic.
Monte Carlo simulations
Design
In this section, {we use simulation experiments to assess the finite
sample performance of the FE-QR and MD-QR estimators discussed previously.}
We consider a simple model as in equation (ref), where
the response $y_{it}$ is generated by
equation*[equation* omitted — 73 chars of source]
This is a general location-scale-shift model. When $\lambda=0$,
the exogenous covariate $x_{it}$ exerts a pure location
shift effect. When $\lambda \neq 0$, $x_{it}$ exerts both location and scale
effects.
We set $\beta=1$ and consider $\lambda\in\{0, 1\}$. We employ three different schemes to generate the disturbances $u_{it}$. Under scheme 1, we generate $u_{it}$ as a $N(0,1)$, under scheme 2 we generate $u_{it}$ as a $t$-distribution with 3 degrees of freedom, and under scheme 3 the disturbances follow $\chi_{3}^{2}$. In this model the true coefficients take the form $\alpha_{i0}(\tau)=\alpha_{i}+F^{-1}(\tau)$ and $\beta_{0}(\tau)=\beta+\lambda F^{-1}(\tau)$ where $F$ denotes the CDF of $u_{it}$.
In all cases, the fixed effects, $\alpha_{i}$, are generated as $\alpha_i = i/n, i=1,...,n$.
The independent variable, $x_{it}$, is generated according to $x_{it}=0.3\alpha_{i}+v_{it}$, where $v_{it}\sim U[0,10]$. This method of generating $\alpha_{i}$ ensures that the classical
random effects estimators are biased because the individual effects and the explanatory variables are correlated.
In the numerical simulations we study the FE-QR defined in (ref), which includes the dummy variables to account for the fixed effects, and the MD-QR estimator in (ref) using the Hendricks-Koenker sandwich covariance matrix estimator in (ref) as weights.\footnote{For estimation of the asymptotic covariance matrix, we use the default bandwidth option in the {\tt quantreg} package in R{; for the Hendricks-Koenker estiamtor this corresponds to the Hall-Sheather method.}} Moreover, for comparison, we report results for the infeasible MD-QR estimator generated with true weights instead of the estimated weights, which are computed by using the true sparsity function in the corresponding variance-covariance matrix. It is important to present results for the MD-QR with the true sparsity to investigate to what extent the finite sample performance of the estimator is affected by estimation of the sparsity function in the weights. The MD-QR using the true sparsity is denoted by MDT-QR.
We report results for bias and standard errors (SE) for the FE-QR, MD-QR, and MDT-QR estimators for the quantiles $\tau=\{0.25,0.5,0.75\}$. All reported results are based on $2,000$ Monte Carlo replications.
Results
Before discussing the Monte Carlo results, it is worthwhile to review the theoretical predictions for the estimators. Remarks (ref) and (ref) in Section (ref) above discussed properties of the MD-QR and FE-QR estimator. The results there provide an upper bound for the order of bias of both estimators which are of the form $(\log T)/T$ and thus depend only on the time series dimension $T$. This indicates that $T\times bias$ should be at most constant or increase very slowly. In addition, the standard error should be proportional to $\sqrt{nT}$.
{To illustrate those theoretical predictions we report $T\times bias$ for the three estimators described above, FE-QR, MD-QR, and MDT-QR. Table (ref), Table (ref), and Table (ref) present results for $\lambda = 0$ for all three estimators and normal, $t_3$ and $\chi_3^2$ errors, respectively. Table (ref), Table (ref), and Table (ref) present the corresponding results for $\lambda = 1$. Moreover, $\sqrt{nT}\times SE$ is reported in Table (ref), Table (ref), and Table (ref) for the location case ($\lambda=0$) for all three estimators and the same three error distributions as above, respectively. Finally, Table (ref), Table (ref), and Table (ref) display the corresponding results to the location-scale ($\lambda=1$) case. }
The first important observation we find when examining Tables (ref), (ref), (ref) is that $T\times bias$ is indeed roughly constant across $n$ for all three estimators considered, indicating that our upper bound is close to being sharp. Tables (ref), (ref), (ref) further indicate that the bias in the pure location-shift model (i.e. $\lambda = 0$) seems to be extremely small even when multiplied by $T$ for all estimators, hinting that in some models the bias might be of even higher order than $T^{-1}$. This does not contradict our theory since we only provide an upper bound for the bias. Another interesting finding is that the exact bias of the three estimators can be quite different (see in particular Table (ref) and Table (ref)). Again, this does not contradict our theoretical predictions since we provide an upper bound with rate while the difference seems to be in the constants. It also seems that estimating the weight in the MD estimator can have a significant impact on the constant in the bias. Finally, there is no estimator that has uniformly smallest bias.
{The scaled (by $\sqrt{nT}$) standard errors are reported in Table (ref), Table (ref), and Table (ref) for the location case and in Table (ref), Table (ref), and Table (ref) for the location-shift case. The first main result is that $\sqrt{nT}\times SE$ is stable across a range of $n,T$ for both $\lambda=0$ and $\lambda=1$, with a slight decreasing trend as $T$ grows, especially for the MD-QR estimator. Both FE-QR and MDT-QR suffer less variation. This slight larger trend for MD-QR is probably due to higher-order terms that result from estimated weights in the MD-QR estimator. In addition, when comparing $\sqrt{nT}\times SE$ for MD-QR and MDT-QR we see that the latter starts with smaller results (for small $T$), however, this difference tends to disappear as $T$ increases.}
figure[figure omitted — 192 chars of source]
{Finally, to further illustrate the discussion in Remark (ref), Figure (ref) shows different plots of the non-standardized bias (dotted lines) and SE (solid lines) as functions of $n$ for different values of $T$ for $\lambda=1$, $\tau=0.75$ and $\chi_3^2$ errors, for the MD-QR estimator. Plots for the FE-QR estimator are similar and we omit them for brevity. As expected from the theory, a comparison for each particular plot reveals that bias is stable across $n$, and a comparison across plots shows that the bias only depends on $T$ while the standard error decreases with $n$ and $T$.\footnote{For robustness purposes, we have conducted experiments with $n=5,000$ and $T=25$ and $T=100$. The results are qualitatively similar, and we omit them for brevity.} Approximately unbiased normality is expected to hold when the standard error is larger than the bias, which is the case when $T \gg n$ in the plots. For a fixed $T$, as $n$ increases, the standard error decreases and at some point is of the same order as the bias; this is when unbiased normality fails. Nevertheless, as predicted this happens for larger and larger values of $n$ as $T$ increases.}
scriptsize\begin{longtable} {ccc|ccc||ccc||ccc}
\caption{$T$ $\times$ bias for different estimators with normal errors. Location-shift ($\lambda=0$)}\\
\\
\hline
& & &\multicolumn{3}{c}{$\tau = 0.25$} & \multicolumn{3}{c}{$\tau = 0.5$} & \multicolumn{3}{c}{$\tau = 0.75$}\\
\hline
$n$ & $T$ & $n/T$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ \\
\hline
25 & 25 & 1.00 & -0.014 & -0.011 & -0.002 & -0.012 & -0.013 & -0.008 & 0.003 & -0.003 & -0.004 \\
25 & 50 & 0.50 & -0.005 & -0.000 & -0.008 & -0.011 & -0.012 & -0.013 & -0.025 & -0.013 & -0.019 \\
25 & 100 & 0.25 & -0.007 & -0.006 & -0.004 & -0.004 & -0.018 & -0.006 & 0.022 & 0.031 & 0.016 \\
25 & 250 & 0.10 & 0.007 & 0.001 & 0.003 & -0.015 & -0.036 & -0.011 & -0.015 & -0.021 & -0.014 \\
50 & 25 & 2.00 & 0.003 & -0.007 & -0.002 & -0.008 & -0.009 & -0.005 & -0.016 & -0.009 & -0.009 \\
50 & 50 & 1.00 & -0.011 & -0.006 & -0.005 & -0.007 & -0.010 & -0.003 & -0.008 & 0.009 & -0.005 \\
50 & 100 & 0.50 & -0.030 & -0.036 & -0.032 & -0.019 & -0.027 & -0.024 & -0.020 & -0.023 & -0.026 \\
50 & 250 & 0.20 & 0.010 & 0.009 & 0.013 & 0.017 & 0.017 & 0.012 & 0.009 & 0.013 & 0.010 \\
100 & 25 & 4.00 & -0.004 & -0.002 & -0.002 & -0.003 & -0.006 & -0.002 & -0.002 & 0.001 & -0.002 \\
100 & 50 & 2.00 & -0.020 & -0.019 & -0.018 & -0.016 & -0.016 & -0.015 & -0.011 & -0.016 & -0.015 \\
100 & 100 & 1.00 & -0.010 & -0.016 & -0.009 & -0.019 & -0.022 & -0.019 & 0.004 & 0.002 & -0.002 \\
100 & 250 & 0.40 & -0.012 & -0.008 & -0.009 & -0.009 & -0.009 & -0.013 & -0.020 & -0.020 & -0.025 \\
250 & 25 & 10.00 & 0.001 & 0.002 & 0.000 & -0.001 & -0.002 & -0.002 & -0.002 & -0.002 & -0.002 \\
250 & 50 & 5.00 & 0.004 & 0.003 & 0.005 & 0.003 & 0.005 & 0.005 & 0.003 & 0.001 & 0.002 \\
250 & 100 & 2.50 & -0.005 & -0.005 & -0.002 & -0.006 & -0.008 & -0.007 & -0.010 & -0.013 & -0.008 \\
250 & 250 & 1.00 & 0.005 & 0.003 & 0.005 & 0.017 & 0.015 & 0.014 & 0.004 & 0.001 & 0.003 \\
\hline\hline
\end{longtable}
scriptsize\begin{longtable} {ccc|ccc||ccc||ccc}
\caption{$\sqrt{nT}$ $\times$ standard error for different estimators with normal errors. Location-shift ($\lambda=0$)}\\
\\
\hline
& & &\multicolumn{3}{c}{$\tau = 0.25$} & \multicolumn{3}{c}{$\tau = 0.5$} & \multicolumn{3}{c}{$\tau = 0.75$}\\
\hline
$n$ & $T$ & $n/T$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ \\
\hline
25 & 25 & 1.00 & 0.541 & 0.485 & 0.480 & 0.467 & 0.445 & 0.438 & 0.547 & 0.495 & 0.489 \\
25 & 50 & 0.50 & 0.514 & 0.482 & 0.484 & 0.456 & 0.449 & 0.444 & 0.527 & 0.479 & 0.481 \\
25 & 100 & 0.25 & 0.490 & 0.481 & 0.474 & 0.452 & 0.451 & 0.442 & 0.497 & 0.479 & 0.471 \\
25 & 250 & 0.10 & 0.488 & 0.477 & 0.480 & 0.452 & 0.448 & 0.445 & 0.493 & 0.474 & 0.476 \\
50 & 25 & 2.00 & 0.554 & 0.487 & 0.490 & 0.468 & 0.441 & 0.440 & 0.570 & 0.489 & 0.491 \\
50 & 50 & 1.00 & 0.511 & 0.477 & 0.477 & 0.464 & 0.456 & 0.449 & 0.497 & 0.477 & 0.469 \\
50 & 100 & 0.50 & 0.499 & 0.476 & 0.483 & 0.449 & 0.439 & 0.438 & 0.503 & 0.477 & 0.480 \\
50 & 250 & 0.20 & 0.478 & 0.467 & 0.467 & 0.440 & 0.443 & 0.437 & 0.480 & 0.475 & 0.470 \\
100 & 25 & 4.00 & 0.554 & 0.481 & 0.479 & 0.476 & 0.449 & 0.444 & 0.552 & 0.479 & 0.484 \\
100 & 50 & 2.00 & 0.504 & 0.473 & 0.467 & 0.461 & 0.444 & 0.446 & 0.523 & 0.473 & 0.480 \\
100 & 100 & 1.00 & 0.492 & 0.475 & 0.471 & 0.446 & 0.437 & 0.438 & 0.502 & 0.478 & 0.478 \\
100 & 250 & 0.40 & 0.477 & 0.463 & 0.468 & 0.427 & 0.422 & 0.423 & 0.480 & 0.465 & 0.471 \\
250 & 25 & 10.00 & 0.567 & 0.495 & 0.489 & 0.472 & 0.450 & 0.445 & 0.568 & 0.487 & 0.484 \\
250 & 50 & 5.00 & 0.507 & 0.474 & 0.466 & 0.468 & 0.455 & 0.450 & 0.513 & 0.476 & 0.480 \\
250 & 100 & 2.50 & 0.489 & 0.475 & 0.476 & 0.437 & 0.424 & 0.426 & 0.491 & 0.477 & 0.471 \\
250 & 250 & 1.00 & 0.478 & 0.470 & 0.469 & 0.443 & 0.438 & 0.437 & 0.476 & 0.469 & 0.462 \\
\hline\hline
\end{longtable}
scriptsize\begin{longtable} {ccc|ccc||ccc||ccc}
\caption{$T$ $\times$ bias for different estimators with $t_3$ errors. Location-shift ($\lambda=0$)}\\
\\
\hline
& & &\multicolumn{3}{c}{$\tau = 0.25$} & \multicolumn{3}{c}{$\tau = 0.5$} & \multicolumn{3}{c}{$\tau = 0.75$}\\
\hline
$n$ & $T$ & $n/T$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ \\
\hline
25 & 25 & 1.00 & -0.004 & 0.003 & -0.008 & -0.010 & -0.005 & -0.001 & -0.023 & -0.015 & -0.011 \\
25 & 50 & 0.50 & 0.006 & -0.006 & -0.009 & 0.008 & -0.010 & 0.011 & 0.002 & 0.003 & 0.004 \\
25 & 100 & 0.25 & 0.020 & 0.008 & 0.013 & 0.009 & -0.006 & 0.003 & -0.011 & -0.026 & -0.025 \\
25 & 250 & 0.10 & 0.065 & 0.019 & 0.037 & -0.044 & -0.024 & -0.026 & -0.016 & -0.036 & -0.012 \\
50 & 25 & 2.00 & -0.000 & -0.001 & -0.009 & 0.003 & 0.000 & 0.003 & -0.020 & -0.003 & 0.003 \\
50 & 50 & 1.00 & -0.005 & 0.001 & 0.008 & 0.001 & 0.002 & 0.004 & -0.005 & -0.016 & -0.006 \\
50 & 100 & 0.50 & -0.005 & 0.000 & -0.010 & -0.008 & -0.000 & -0.007 & -0.021 & -0.009 & -0.009 \\
50 & 250 & 0.20 & -0.025 & -0.019 & -0.031 & 0.018 & 0.029 & 0.023 & 0.009 & -0.008 & 0.005 \\
100 & 25 & 4.00 & -0.003 & -0.001 & -0.007 & 0.001 & -0.002 & 0.001 & -0.004 & -0.008 & -0.006 \\
100 & 50 & 2.00 & -0.003 & -0.004 & -0.003 & 0.001 & -0.000 & -0.001 & 0.001 & -0.002 & 0.000 \\
100 & 100 & 1.00 & 0.001 & -0.004 & -0.009 & -0.015 & -0.012 & -0.019 & 0.008 & 0.000 & -0.003 \\
100 & 250 & 0.40 & 0.011 & 0.004 & 0.006 & 0.015 & 0.003 & 0.011 & -0.009 & -0.008 & -0.009 \\
250 & 25 & 10.00 & 0.003 & 0.000 & 0.002 & -0.003 & -0.002 & -0.001 & -0.005 & -0.004 & -0.002 \\
250 & 50 & 5.00 & -0.003 & -0.007 & -0.004 & -0.003 & 0.002 & -0.001 & 0.005 & 0.003 & 0.002 \\
250 & 100 & 2.50 & 0.003 & -0.003 & 0.004 & 0.008 & 0.004 & 0.004 & -0.010 & -0.007 & -0.008 \\
250 & 250 & 1.00 & -0.002 & -0.001 & 0.002 & 0.004 & 0.005 & 0.004 & 0.018 & 0.018 & 0.012 \\
\hline\hline
\end{longtable}
scriptsize\begin{longtable} {ccc|ccc||ccc||ccc}
\caption{$\sqrt{nT}$ $\times$ standard error for different estimators with $t_3$ errors. Location-shift ($\lambda=0$)}\\
\\
\hline
& & &\multicolumn{3}{c}{$\tau = 0.25$} & \multicolumn{3}{c}{$\tau = 0.5$} & \multicolumn{3}{c}{$\tau = 0.75$}\\
\hline
$n$ & $T$ & $n/T$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ \\
\hline
25 & 25 & 1.00 & 0.738 & 0.611 & 0.642 & 0.546 & 0.499 & 0.519 & 0.750 & 0.613 & 0.651 \\
25 & 50 & 0.50 & 0.659 & 0.604 & 0.600 & 0.506 & 0.489 & 0.495 & 0.645 & 0.583 & 0.604 \\
25 & 100 & 0.25 & 0.614 & 0.586 & 0.592 & 0.480 & 0.471 & 0.477 & 0.604 & 0.582 & 0.577 \\
25 & 250 & 0.10 & 0.604 & 0.595 & 0.590 & 0.473 & 0.472 & 0.466 & 0.592 & 0.575 & 0.580 \\
50 & 25 & 2.00 & 0.756 & 0.603 & 0.632 & 0.524 & 0.487 & 0.498 & 0.724 & 0.593 & 0.617 \\
50 & 50 & 1.00 & 0.656 & 0.592 & 0.600 & 0.493 & 0.485 & 0.488 & 0.646 & 0.585 & 0.592 \\
50 & 100 & 0.50 & 0.607 & 0.578 & 0.582 & 0.471 & 0.464 & 0.464 & 0.609 & 0.590 & 0.594 \\
50 & 250 & 0.20 & 0.603 & 0.589 & 0.594 & 0.487 & 0.476 & 0.480 & 0.606 & 0.596 & 0.594 \\
100 & 25 & 4.00 & 0.738 & 0.593 & 0.611 & 0.522 & 0.481 & 0.502 & 0.737 & 0.584 & 0.619 \\
100 & 50 & 2.00 & 0.653 & 0.584 & 0.602 & 0.489 & 0.472 & 0.477 & 0.649 & 0.581 & 0.602 \\
100 & 100 & 1.00 & 0.630 & 0.588 & 0.605 & 0.498 & 0.483 & 0.486 & 0.625 & 0.592 & 0.604 \\
100 & 250 & 0.40 & 0.602 & 0.590 & 0.593 & 0.482 & 0.478 & 0.478 & 0.603 & 0.585 & 0.588 \\
250 & 25 & 10.00 & 0.749 & 0.596 & 0.640 & 0.539 & 0.479 & 0.500 & 0.746 & 0.582 & 0.614 \\
250 & 50 & 5.00 & 0.676 & 0.597 & 0.601 & 0.505 & 0.484 & 0.491 & 0.660 & 0.602 & 0.610 \\
250 & 100 & 2.50 & 0.624 & 0.592 & 0.603 & 0.493 & 0.482 & 0.487 & 0.620 & 0.593 & 0.598 \\
250 & 250 & 1.00 & 0.593 & 0.572 & 0.581 & 0.466 & 0.458 & 0.460 & 0.604 & 0.589 & 0.590 \\
\hline\hline
\end{longtable}
scriptsize\begin{longtable} {ccc|ccc||ccc||ccc}
\caption{$T$ $\times$ bias for different estimators with $\chi_3^2$ errors. Location-shift ($\lambda=0$)}\\
\\
\hline
& & &\multicolumn{3}{c}{$\tau = 0.25$} & \multicolumn{3}{c}{$\tau = 0.5$} & \multicolumn{3}{c}{$\tau = 0.75$}\\
\hline
$n$ & $T$ & $n/T$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ \\
\hline
25 & 25 & 1.00 & 0.004 & 0.017 & 0.005 & 0.014 & 0.001 & 0.019 & 0.009 & 0.014 & -0.009 \\
25 & 50 & 0.50 & -0.017 & -0.019 & -0.016 & -0.019 & -0.025 & -0.014 & -0.048 & -0.017 & -0.057 \\
25 & 100 & 0.25 & -0.015 & -0.030 & -0.007 & -0.010 & -0.018 & -0.021 & -0.069 & -0.077 & -0.084 \\
25 & 250 & 0.10 & -0.030 & -0.031 & -0.025 & -0.023 & -0.024 & -0.026 & 0.024 & 0.013 & 0.015 \\
50 & 25 & 2.00 & -0.005 & -0.011 & -0.012 & -0.006 & -0.015 & -0.006 & -0.035 & -0.008 & -0.024 \\
50 & 50 & 1.00 & -0.010 & -0.005 & -0.012 & -0.007 & -0.002 & -0.014 & -0.010 & -0.036 & -0.038 \\
50 & 100 & 0.50 & -0.008 & -0.008 & 0.004 & 0.037 & 0.013 & 0.019 & 0.058 & 0.042 & 0.040 \\
50 & 250 & 0.20 & -0.011 & 0.006 & -0.014 & -0.020 & -0.003 & -0.022 & 0.129 & 0.131 & 0.124 \\
100 & 25 & 4.00 & -0.006 & -0.010 & -0.000 & -0.001 & -0.001 & -0.010 & -0.018 & -0.018 & -0.024 \\
100 & 50 & 2.00 & -0.006 & -0.012 & -0.003 & -0.003 & 0.002 & -0.000 & 0.022 & 0.029 & 0.022 \\
100 & 100 & 1.00 & -0.001 & 0.008 & 0.007 & 0.006 & 0.011 & 0.010 & 0.022 & 0.009 & 0.009 \\
100 & 250 & 0.40 & 0.003 & -0.000 & 0.006 & 0.023 & 0.030 & 0.030 & 0.047 & 0.068 & 0.058 \\
250 & 25 & 10.00 & -0.002 & 0.000 & 0.002 & -0.007 & -0.007 & 0.002 & -0.017 & -0.009 & -0.012 \\
250 & 50 & 5.00 & 0.003 & 0.004 & 0.006 & -0.003 & -0.005 & 0.002 & 0.013 & 0.017 & 0.009 \\
250 & 100 & 2.50 & 0.003 & -0.002 & 0.002 & 0.021 & 0.011 & 0.018 & 0.040 & 0.036 & 0.034 \\
250 & 250 & 1.00 & -0.009 & -0.007 & -0.009 & -0.040 & -0.032 & -0.033 & 0.010 & 0.018 & 0.014 \\
\hline\hline
\end{longtable}
scriptsize\begin{longtable} {ccc|ccc||ccc||ccc}
\caption{$\sqrt{nT}$ $\times$ standard error for different estimators with $\chi_3^2$ errors. Location-shift ($\lambda=0$)}\\
\\
\hline
& & &\multicolumn{3}{c}{$\tau = 0.25$} & \multicolumn{3}{c}{$\tau = 0.5$} & \multicolumn{3}{c}{$\tau = 0.75$}\\
\hline
$n$ & $T$ & $n/T$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ \\
\hline
25 & 25 & 1.00 & 0.664 & 0.636 & 0.650 & 0.972 & 0.938 & 0.943 & 1.668 & 1.446 & 1.473 \\
25 & 50 & 0.50 & 0.646 & 0.618 & 0.623 & 0.954 & 0.935 & 0.939 & 1.617 & 1.459 & 1.471 \\
25 & 100 & 0.25 & 0.638 & 0.627 & 0.633 & 0.963 & 0.943 & 0.947 & 1.553 & 1.484 & 1.498 \\
25 & 250 & 0.10 & 0.630 & 0.625 & 0.622 & 0.924 & 0.918 & 0.916 & 1.496 & 1.456 & 1.466 \\
50 & 25 & 2.00 & 0.658 & 0.624 & 0.628 & 0.987 & 0.919 & 0.933 & 1.726 & 1.455 & 1.496 \\
50 & 50 & 1.00 & 0.634 & 0.617 & 0.625 & 0.977 & 0.953 & 0.962 & 1.619 & 1.462 & 1.472 \\
50 & 100 & 0.50 & 0.630 & 0.613 & 0.617 & 0.939 & 0.913 & 0.918 & 1.515 & 1.450 & 1.456 \\
50 & 250 & 0.20 & 0.655 & 0.633 & 0.641 & 0.946 & 0.946 & 0.943 & 1.504 & 1.462 & 1.470 \\
100 & 25 & 4.00 & 0.663 & 0.617 & 0.637 & 1.023 & 0.943 & 0.958 & 1.739 & 1.485 & 1.493 \\
100 & 50 & 2.00 & 0.647 & 0.615 & 0.625 & 0.949 & 0.915 & 0.927 & 1.603 & 1.438 & 1.440 \\
100 & 100 & 1.00 & 0.643 & 0.626 & 0.632 & 0.945 & 0.929 & 0.926 & 1.549 & 1.455 & 1.477 \\
100 & 250 & 0.40 & 0.642 & 0.627 & 0.636 & 0.966 & 0.940 & 0.954 & 1.508 & 1.453 & 1.460 \\
250 & 25 & 10.00 & 0.671 & 0.619 & 0.631 & 0.968 & 0.924 & 0.920 & 1.745 & 1.456 & 1.462 \\
250 & 50 & 5.00 & 0.661 & 0.631 & 0.646 & 0.974 & 0.929 & 0.954 & 1.586 & 1.439 & 1.462 \\
250 & 100 & 2.50 & 0.649 & 0.633 & 0.640 & 0.972 & 0.932 & 0.946 & 1.579 & 1.462 & 1.475 \\
250 & 250 & 1.00 & 0.644 & 0.630 & 0.632 & 0.923 & 0.911 & 0.911 & 1.450 & 1.434 & 1.419 \\
\hline\hline
\end{longtable}
scriptsize\begin{longtable} {ccc|ccc||ccc||ccc}
\caption{$T$ $\times$ bias for different estimators with normal errors. Location-scale shift ($\lambda=1$)}\\
\\
\hline
& & &\multicolumn{3}{c}{$\tau = 0.25$} & \multicolumn{3}{c}{$\tau = 0.5$} & \multicolumn{3}{c}{$\tau = 0.75$}\\
\hline
$n$ & $T$ & $n/T$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ \\
\hline
25 & 25 & 1.00 & 0.992 & 0.867 & 0.948 & -0.038 & -0.054 & -0.006 & -0.993 & -0.830 & -0.890 \\
25 & 50 & 0.50 & 1.151 & 0.870 & 0.876 & -0.077 & -0.095 & -0.095 & -1.167 & -0.886 & -0.943 \\
25 & 100 & 0.25 & 1.171 & 0.799 & 0.843 & -0.093 & -0.107 & -0.140 & -1.051 & -0.750 & -0.788 \\
25 & 250 & 0.10 & 1.514 & 0.969 & 1.041 & -0.069 & -0.152 & -0.055 & -1.354 & -0.894 & -0.877 \\
50 & 25 & 2.00 & 1.089 & 0.879 & 0.914 & -0.043 & -0.034 & -0.060 & -1.158 & -0.870 & -1.005 \\
50 & 50 & 1.00 & 1.073 & 0.816 & 0.895 & -0.048 & -0.026 & -0.039 & -1.094 & -0.810 & -0.926 \\
50 & 100 & 0.50 & 1.091 & 0.648 & 0.767 & -0.112 & -0.137 & -0.140 & -1.284 & -0.935 & -0.974 \\
50 & 250 & 0.20 & 1.411 & 0.947 & 0.949 & 0.029 & -0.013 & 0.012 & -1.269 & -0.859 & -0.805 \\
100 & 25 & 4.00 & 1.114 & 0.857 & 0.916 & -0.002 & -0.018 & -0.004 & -1.075 & -0.826 & -0.925 \\
100 & 50 & 2.00 & 1.015 & 0.777 & 0.821 & -0.082 & -0.063 & -0.076 & -1.147 & -0.930 & -0.960 \\
100 & 100 & 1.00 & 1.169 & 0.754 & 0.834 & -0.141 & -0.134 & -0.143 & -1.222 & -0.868 & -0.909 \\
100 & 250 & 0.40 & 1.314 & 0.783 & 0.849 & -0.075 & -0.071 & -0.069 & -1.396 & -0.930 & -0.926 \\
250 & 25 & 10.00 & 1.106 & 0.890 & 0.923 & -0.001 & -0.017 & -0.014 & -1.109 & -0.879 & -0.936 \\
250 & 50 & 5.00 & 1.127 & 0.886 & 0.934 & 0.009 & 0.011 & 0.013 & -1.071 & -0.854 & -0.887 \\
250 & 100 & 2.50 & 1.161 & 0.805 & 0.871 & -0.038 & -0.031 & -0.038 & -1.232 & -0.884 & -0.892 \\
250 & 250 & 1.00 & 1.351 & 0.802 & 0.870 & 0.027 & 0.000 & 0.031 & -1.335 & -0.838 & -0.861 \\
\hline\hline
\end{longtable}
scriptsize\begin{longtable} {ccc|ccc||ccc||ccc}
\caption{$\sqrt{nT}$ $\times$ standard error for different estimators with normal errors. Location-scale shift ($\lambda=1$)}\\
\\
\hline
& & &\multicolumn{3}{c}{$\tau = 0.25$} & \multicolumn{3}{c}{$\tau = 0.5$} & \multicolumn{3}{c}{$\tau = 0.75$}\\
\hline
$n$ & $T$ & $n/T$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ \\
\hline
25 & 25 & 1.00 & 3.222 & 2.791 & 2.841 & 2.661 & 2.572 & 2.526 & 3.186 & 2.857 & 2.870 \\
25 & 50 & 0.50 & 2.909 & 2.744 & 2.739 & 2.574 & 2.528 & 2.487 & 2.963 & 2.750 & 2.757 \\
25 & 100 & 0.25 & 2.771 & 2.688 & 2.701 & 2.511 & 2.462 & 2.469 & 2.736 & 2.658 & 2.634 \\
25 & 250 & 0.10 & 2.707 & 2.646 & 2.669 & 2.462 & 2.427 & 2.438 & 2.725 & 2.628 & 2.628 \\
50 & 25 & 2.00 & 3.195 & 2.806 & 2.853 & 2.655 & 2.533 & 2.507 & 3.203 & 2.837 & 2.810 \\
50 & 50 & 1.00 & 2.934 & 2.733 & 2.699 & 2.608 & 2.547 & 2.520 & 2.832 & 2.702 & 2.686 \\
50 & 100 & 0.50 & 2.788 & 2.667 & 2.680 & 2.493 & 2.448 & 2.430 & 2.782 & 2.664 & 2.701 \\
50 & 250 & 0.20 & 2.694 & 2.644 & 2.633 & 2.510 & 2.474 & 2.479 & 2.711 & 2.656 & 2.661 \\
100 & 25 & 4.00 & 3.332 & 2.799 & 2.813 & 2.703 & 2.574 & 2.566 & 3.233 & 2.774 & 2.837 \\
100 & 50 & 2.00 & 2.853 & 2.682 & 2.678 & 2.570 & 2.477 & 2.480 & 2.960 & 2.703 & 2.740 \\
100 & 100 & 1.00 & 2.800 & 2.712 & 2.674 & 2.494 & 2.434 & 2.440 & 2.749 & 2.652 & 2.628 \\
100 & 250 & 0.40 & 2.706 & 2.678 & 2.660 & 2.375 & 2.358 & 2.360 & 2.655 & 2.601 & 2.599 \\
250 & 25 & 10.00 & 3.302 & 2.833 & 2.860 & 2.675 & 2.504 & 2.533 & 3.398 & 2.809 & 2.847 \\
250 & 50 & 5.00 & 2.948 & 2.714 & 2.736 & 2.678 & 2.580 & 2.552 & 2.926 & 2.736 & 2.770 \\
250 & 100 & 2.50 & 2.788 & 2.709 & 2.706 & 2.421 & 2.342 & 2.370 & 2.728 & 2.644 & 2.641 \\
250 & 250 & 1.00 & 2.695 & 2.652 & 2.638 & 2.460 & 2.441 & 2.434 & 2.676 & 2.650 & 2.627 \\
\hline\hline
\end{longtable}
scriptsize\begin{longtable} {ccc|ccc||ccc||ccc}
\caption{$T$ $\times$ bias for different estimators with $t_3$ errors. Location-scale shift ($\lambda=1$)}\\
\\
\hline
& & &\multicolumn{3}{c}{$\tau = 0.25$} & \multicolumn{3}{c}{$\tau = 0.5$} & \multicolumn{3}{c}{$\tau = 0.75$}\\
\hline
$n$ & $T$ & $n/T$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ \\
\hline
25 & 25 & 1.00 & 2.924 & 1.424 & 0.781 & -0.032 & -0.026 & -0.016 & -3.020 & -1.435 & -0.849 \\
25 & 50 & 0.50 & 2.898 & 1.416 & 0.795 & 0.112 & 0.044 & 0.109 & -2.824 & -1.368 & -0.774 \\
25 & 100 & 0.25 & 2.882 & 1.460 & 0.837 & 0.006 & -0.037 & 0.014 & -2.912 & -1.571 & -1.002 \\
25 & 250 & 0.10 & 3.374 & 1.826 & 1.127 & -0.086 & -0.069 & -0.007 & -3.161 & -1.788 & -1.032 \\
50 & 25 & 2.00 & 2.991 & 1.380 & 0.746 & 0.024 & 0.033 & 0.057 & -3.043 & -1.396 & -0.737 \\
50 & 50 & 1.00 & 2.934 & 1.462 & 0.906 & -0.004 & -0.022 & 0.005 & -2.943 & -1.490 & -0.890 \\
50 & 100 & 0.50 & 3.019 & 1.544 & 0.909 & -0.025 & 0.056 & -0.017 & -3.015 & -1.519 & -0.899 \\
50 & 250 & 0.20 & 2.999 & 1.404 & 0.810 & 0.107 & 0.211 & 0.137 & -3.006 & -1.593 & -0.891 \\
100 & 25 & 4.00 & 2.926 & 1.419 & 0.746 & -0.016 & -0.003 & -0.004 & -3.053 & -1.434 & -0.803 \\
100 & 50 & 2.00 & 3.015 & 1.494 & 0.856 & 0.054 & 0.029 & 0.023 & -2.974 & -1.511 & -0.855 \\
100 & 100 & 1.00 & 3.021 & 1.517 & 0.855 & 0.013 & -0.013 & -0.021 & -2.912 & -1.510 & -0.897 \\
100 & 250 & 0.40 & 3.070 & 1.500 & 0.885 & 0.083 & 0.072 & 0.058 & -3.046 & -1.563 & -0.899 \\
250 & 25 & 10.00 & 3.039 & 1.439 & 0.793 & -0.002 & -0.007 & 0.000 & -3.057 & -1.453 & -0.824 \\
250 & 50 & 5.00 & 2.945 & 1.472 & 0.830 & 0.002 & 0.035 & 0.007 & -2.944 & -1.493 & -0.833 \\
250 & 100 & 2.50 & 2.989 & 1.466 & 0.869 & 0.045 & 0.027 & 0.024 & -3.041 & -1.522 & -0.901 \\
250 & 250 & 1.00 & 3.071 & 1.516 & 0.918 & 0.051 & 0.036 & 0.055 & -2.986 & -1.389 & -0.803 \\
\hline\hline
\end{longtable}
scriptsize\begin{longtable} {ccc|ccc||ccc||ccc}
\caption{$\sqrt{nT}$ $\times$ standard error for different estimators with $t_3$ errors. Location-scale shift ($\lambda=1$)}\\
\\
\hline
& & &\multicolumn{3}{c}{$\tau = 0.25$} & \multicolumn{3}{c}{$\tau = 0.5$} & \multicolumn{3}{c}{$\tau = 0.75$}\\
\hline
$n$ & $T$ & $n/T$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ \\
\hline
25 & 25 & 1.00 & 4.274 & 3.377 & 3.666 & 3.080 & 2.876 & 2.953 & 4.452 & 3.515 & 3.757 \\
25 & 50 & 0.50 & 3.715 & 3.373 & 3.436 & 2.860 & 2.737 & 2.798 & 3.611 & 3.300 & 3.406 \\
25 & 100 & 0.25 & 3.442 & 3.238 & 3.325 & 2.686 & 2.666 & 2.679 & 3.464 & 3.275 & 3.280 \\
25 & 250 & 0.10 & 3.403 & 3.321 & 3.318 & 2.668 & 2.632 & 2.652 & 3.310 & 3.216 & 3.279 \\
50 & 25 & 2.00 & 4.371 & 3.433 & 3.620 & 3.025 & 2.792 & 2.899 & 4.243 & 3.326 & 3.574 \\
50 & 50 & 1.00 & 3.681 & 3.329 & 3.364 & 2.774 & 2.690 & 2.750 & 3.728 & 3.336 & 3.397 \\
50 & 100 & 0.50 & 3.475 & 3.291 & 3.317 & 2.663 & 2.659 & 2.656 & 3.452 & 3.321 & 3.327 \\
50 & 250 & 0.20 & 3.367 & 3.294 & 3.356 & 2.706 & 2.670 & 2.684 & 3.366 & 3.290 & 3.266 \\
100 & 25 & 4.00 & 4.342 & 3.412 & 3.585 & 3.004 & 2.777 & 2.878 & 4.446 & 3.368 & 3.616 \\
100 & 50 & 2.00 & 3.717 & 3.337 & 3.428 & 2.804 & 2.668 & 2.747 & 3.728 & 3.300 & 3.444 \\
100 & 100 & 1.00 & 3.505 & 3.296 & 3.347 & 2.766 & 2.677 & 2.717 & 3.561 & 3.397 & 3.394 \\
100 & 250 & 0.40 & 3.346 & 3.208 & 3.277 & 2.695 & 2.652 & 2.679 & 3.389 & 3.313 & 3.323 \\
250 & 25 & 10.00 & 4.418 & 3.419 & 3.686 & 3.064 & 2.790 & 2.869 & 4.439 & 3.335 & 3.574 \\
250 & 50 & 5.00 & 3.782 & 3.363 & 3.422 & 2.793 & 2.762 & 2.744 & 3.724 & 3.342 & 3.431 \\
250 & 100 & 2.50 & 3.482 & 3.272 & 3.346 & 2.772 & 2.716 & 2.746 & 3.501 & 3.364 & 3.397 \\
250 & 250 & 1.00 & 3.319 & 3.249 & 3.273 & 2.571 & 2.551 & 2.546 & 3.354 & 3.290 & 3.300 \\
\hline\hline
\end{longtable}
scriptsize\begin{longtable} {ccc|ccc||ccc||ccc}
\caption{$T$ $\times$ bias for different estimators with $\chi_3^2$ errors. Location-scale shift ($\lambda=1$)}\\
\\
\hline
& & &\multicolumn{3}{c}{$\tau = 0.25$} & \multicolumn{3}{c}{$\tau = 0.5$} & \multicolumn{3}{c}{$\tau = 0.75$}\\
\hline
$n$ & $T$ & $n/T$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ \\
\hline
25 & 25 & 1.00 & -1.428 & 0.389 & 2.106 & -2.447 & -1.072 & 0.686 & -6.874 & -3.783 & -1.770 \\
25 & 50 & 0.50 & -1.533 & 0.098 & 1.720 & -2.682 & -1.259 & 0.371 & -7.136 & -3.987 & -2.302 \\
25 & 100 & 0.25 & -1.207 & 0.021 & 1.641 & -2.614 & -1.136 & 0.404 & -7.732 & -4.135 & -2.344 \\
25 & 250 & 0.10 & -1.259 & -0.021 & 1.369 & -2.988 & -1.304 & 0.189 & -8.085 & -4.153 & -2.283 \\
50 & 25 & 2.00 & -1.625 & 0.227 & 2.026 & -2.640 & -1.158 & 0.558 & -7.454 & -3.987 & -2.022 \\
50 & 50 & 1.00 & -1.630 & 0.136 & 1.719 & -2.703 & -1.185 & 0.388 & -7.373 & -4.137 & -2.220 \\
50 & 100 & 0.50 & -1.507 & -0.069 & 1.525 & -2.719 & -1.098 & 0.403 & -7.607 & -3.890 & -2.112 \\
50 & 250 & 0.20 & -1.307 & -0.009 & 1.346 & -3.107 & -1.237 & 0.164 & -7.682 & -3.618 & -1.732 \\
100 & 25 & 4.00 & -1.690 & 0.244 & 2.050 & -2.564 & -1.081 & 0.655 & -7.461 & -4.050 & -2.005 \\
100 & 50 & 2.00 & -1.655 & 0.041 & 1.734 & -2.772 & -1.175 & 0.404 & -7.314 & -4.144 & -2.068 \\
100 & 100 & 1.00 & -1.501 & -0.006 & 1.556 & -2.839 & -1.221 & 0.383 & -7.632 & -4.104 & -2.161 \\
100 & 250 & 0.40 & -1.294 & -0.117 & 1.388 & -2.964 & -1.202 & 0.316 & -8.206 & -4.126 & -2.003 \\
250 & 25 & 10.00 & -1.672 & 0.216 & 2.084 & -2.667 & -1.128 & 0.648 & -7.533 & -4.131 & -1.930 \\
250 & 50 & 5.00 & -1.658 & 0.055 & 1.759 & -2.825 & -1.192 & 0.417 & -7.388 & -4.127 & -2.053 \\
250 & 100 & 2.50 & -1.519 & -0.091 & 1.510 & -2.830 & -1.186 & 0.363 & -7.705 & -4.088 & -2.132 \\
250 & 250 & 1.00 & -1.357 & -0.129 & 1.351 & -3.220 & -1.393 & 0.117 & -8.269 & -4.215 & -2.104 \\
\hline\hline
\end{longtable}
scriptsize\begin{longtable} {ccc|ccc||ccc||ccc}
\caption{$\sqrt{nT}$ $\times$ standard error for different estimators with $\chi_3^2$ errors. Location-shift ($\lambda=1$)}\\
\\
\hline
& & &\multicolumn{3}{c}{$\tau = 0.25$} & \multicolumn{3}{c}{$\tau = 0.5$} & \multicolumn{3}{c}{$\tau = 0.75$}\\
\hline
$n$ & $T$ & $n/T$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ & $MD$ & $FE$ & $MDT$ \\
\hline
25 & 25 & 1.00 & 3.897 & 3.725 & 3.785 & 5.721 & 5.315 & 5.442 & 9.980 & 8.193 & 8.384 \\
25 & 50 & 0.50 & 3.772 & 3.605 & 3.634 & 5.495 & 5.297 & 5.376 & 8.905 & 8.195 & 8.209 \\
25 & 100 & 0.25 & 3.533 & 3.502 & 3.496 & 5.306 & 5.272 & 5.233 & 8.666 & 8.289 & 8.338 \\
25 & 250 & 0.10 & 3.554 & 3.489 & 3.494 & 5.163 & 5.094 & 5.103 & 8.342 & 8.083 & 7.990 \\
50 & 25 & 2.00 & 4.025 & 3.720 & 3.774 & 5.774 & 5.341 & 5.422 & 10.229 & 8.447 & 8.542 \\
50 & 50 & 1.00 & 3.614 & 3.525 & 3.580 & 5.545 & 5.310 & 5.420 & 9.110 & 8.261 & 8.378 \\
50 & 100 & 0.50 & 3.593 & 3.470 & 3.484 & 5.264 & 5.202 & 5.122 & 8.590 & 8.190 & 8.143 \\
50 & 250 & 0.20 & 3.691 & 3.563 & 3.632 & 5.379 & 5.341 & 5.322 & 8.461 & 8.224 & 8.247 \\
100 & 25 & 4.00 & 3.999 & 3.691 & 3.842 & 5.807 & 5.402 & 5.577 & 10.543 & 8.356 & 8.641 \\
100 & 50 & 2.00 & 3.744 & 3.515 & 3.591 & 5.431 & 5.239 & 5.224 & 8.965 & 8.142 & 8.233 \\
100 & 100 & 1.00 & 3.597 & 3.474 & 3.507 & 5.302 & 5.148 & 5.155 & 8.565 & 8.183 & 8.200 \\
100 & 250 & 0.40 & 3.633 & 3.568 & 3.601 & 5.384 & 5.201 & 5.298 & 8.389 & 8.179 & 8.200 \\
250 & 25 & 10.00 & 4.141 & 3.744 & 3.878 & 5.698 & 5.331 & 5.396 & 10.486 & 8.288 & 8.432 \\
250 & 50 & 5.00 & 3.788 & 3.677 & 3.699 & 5.471 & 5.417 & 5.409 & 9.204 & 8.187 & 8.347 \\
250 & 100 & 2.50 & 3.687 & 3.602 & 3.671 & 5.370 & 5.250 & 5.273 & 8.828 & 8.198 & 8.377 \\
250 & 250 & 1.00 & 3.578 & 3.503 & 3.532 & 5.120 & 5.070 & 5.023 & 8.065 & 7.949 & 7.917 \\
\hline\hline
\end{longtable}
Conclusion
Asymptotic theory for panel data quantile regression (QR) with fixed effects poses many challenges as it involves models with an increasing number of parameters and a non-smooth objective function. Owing to those difficulties, unbiased asymptotic normality of estimators for common parameters in panel data QR has so far only been known to hold under stringent conditions on the length of panels relative to the number of individuals. Specifically, KatoGalvaoMontes-Rojas12 proved $\sqrt{nT}$-consistency of a panel data fixed effect QR (FE-QR) estimator under the stringent condition that $n^{2}(\log n)^{3}/T\rightarrow0$, and since then, it has been an open question whether the rates on the sample size requirement could be improved to a condition which is closer to the assumption $n/T = o(1)$ which is known to be sufficient in nonlinear panel data models with smooth objective function.
The major contribution of this paper was to show that, for both the FE-QR and MD-QR estimators, such an improvement is indeed possible in a wide range of scenarios including observations with temporal dependence.
Our results are important to practitioners and theorists. The main practical implication is that, despite a lack of smoothness of the objective function, panel data QR is applicable for the same type of panel dimensions as other popular nonlinear models. This validates the use of QR in many practical scenarios in which its validity previously lacked theoretical justification. Our theory also provides grounds for subsequent methodological research on panel data QR which now can rely on an improved and more realistic growth rate on the sample size. We believe that the proof techniques provided here will also be useful to future researchers.
There are ample directions for future research. For instance, we have not considered censored observations or scenarios with endogeneity. Another topic that merits further exploration is related to the bootstrap and its refinements.
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